A074377 -id:A074377 - cp's OEIS frontend

A274757 Numbers k such that 6*k+1 is a triangular number (A000217).

0, 9, 15, 42, 54, 99, 117, 180, 204, 285, 315, 414, 450, 567, 609, 744, 792, 945, 999, 1170, 1230, 1419, 1485, 1692, 1764, 1989, 2067, 2310, 2394, 2655, 2745, 3024, 3120, 3417, 3519, 3834, 3942, 4275, 4389, 4740, 4860, 5229, 5355, 5742, 5874, 6279, 6417

Offset: 1

Colin Barker, Jul 04 2016

Numbers of the type floor(3*m*(m+1)/4) for which floor(3*m*(m+1)/4) = 3*floor(m*(m+1)/4). A014601 lists the values of m. - Bruno Berselli, Jan 13 2017

Numbers of the form 3*k*(4*k + 1) for k in Z. - Peter Bala, Nov 21 2024

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A000217, A014601.
Cf. A000096 (k+1), A074377 (2*k+1), A045943 (3*k+1), A274681 (4*k+1), A085787 (5*k+1).
Cf. similar sequences listed in A274830.

Table[3 (2 n - 1) (2 n + (-1)^n - 1)/4, {n, 1, 60}] (* Bruno Berselli, Jul 08 2016 *)
LinearRecurrence[{1,2,-2,-1,1},{0,9,15,42,54},50] (* Harvey P. Dale, Apr 13 2025 *)

PARI
```
isok(n) = ispolygonal(6*n+1, 3)
```

select(n->ispolygonal(6*n+1, 3), vector(7000, n, n-1))

concat(0, Vec(3*x^2*(3+2*x+3*x^2)/((1-x)^3*(1+x)^2) + O(x^60)))

G.f.: 3*x^2*(3 + 2*x + 3*x^2) / ((1 - x)^3*(1 + x)^2).
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5) for n>5.
a(n) = 3*(2*n - 1)*(2*n + (-1)^n - 1)/4. Therefore:
a(n) = 3*n*(2*n - 1)/2 for n even,
a(n) = 3*(n-1)*(2*n - 1)/2 for n odd.

A274830 Numbers k such that 7*k+1 is a triangular number (A000217).

Original entry on oeis.org

0, 2, 5, 11, 17, 27, 36, 50, 62, 80, 95, 117, 135, 161, 182, 212, 236, 270, 297, 335, 365, 407, 440, 486, 522, 572, 611, 665, 707, 765, 810, 872, 920, 986, 1037, 1107, 1161, 1235, 1292, 1370, 1430, 1512, 1575, 1661, 1727, 1817, 1886, 1980, 2052, 2150, 2225

Offset: 1

Colin Barker, Jul 08 2016

From Peter Bala, Nov 21 2024: (Start)
Numbers of the form n*(7*n + 3)/2 for n in Z. Cf. A057570.
The sequence terms occur as the exponents in the expansion of Product_{n >= 1} (1 - x^(7*n)) * (1 + x^(7*n-2)) * (1 + x^(7*n-5)) = 1 + x^2 + x^5 + x^11 + x^17 + x^27 + x^36 + .... Cf. A363800. (End)

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A000217, A057570, A363800.

Cf. similar sequences where k*n+1 is a triangular number: A000096 (k=1), A074377 (k=2), A045943 (k=3), A274681 (k=4), A085787 (k=5), A274757 (k=6).

Table[(14 (n - 1) n + (2 n - 1) (-1)^n + 1)/16, {n, 1, 60}] (* Bruno Berselli, Jul 08 2016 *)

select(n->ispolygonal(7*n+1, 3), vector(3000, n, n-1))

concat(0, Vec(x^2*(2+3*x+2*x^2)/((1-x)^3*(1+x)^2) + O(x^100)))

G.f.: x^2*(2 + 3*x + 2*x^2) / ((1 - x)^3*(1 + x)^2).
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5) for n>5.
a(n) = (14*(n - 1)*n + (2*n - 1)*(-1)^n + 1)/16. Therefore:
a(n) = n*(7*n - 6)/8 for n even,
a(n) = (n - 1)*(7*n - 1)/8 for n odd.
E.g.f.: (x*(7*x -1)*cosh(x) + (7*x^2 + x + 1)*sinh(x))/8. - Stefano Spezia, Nov 26 2024

Edited by Bruno Berselli, Jul 08 2016

A303301 Square array T(n,k) read by antidiagonals upwards in which row n is obtained by taking the general formula for generalized n-gonal numbers: m((n - 2)m - n + 4)/2, where m = 0, +1, -1, +2, -2, +3, -3, ... and n >= 5. Here n >= 0.

Original entry on oeis.org

0, 0, 1, 0, 1, -3, 0, 1, -2, 0, 0, 1, -1, 1, -8, 0, 1, 0, 2, -5, -3, 0, 1, 1, 3, -2, 0, -15, 0, 1, 2, 4, 1, 3, -9, -8, 0, 1, 3, 5, 4, 6, -3, -2, -24, 0, 1, 4, 6, 7, 9, 3, 4, -14, -15, 0, 1, 5, 7, 10, 12, 9, 10, -4, -5, -35, 0, 1, 6, 8, 13, 15, 15, 16, 6, 5, -20, -24, 0, 1, 7, 9, 16, 18, 21, 22, 16, 15, -5, -9, -48

Offset: 0

Omar E. Pol, Jun 08 2018

Note that the formula mentioned in the definition gives several kinds of numbers, for example:
Row 0 and row 1 give A317300 and A317301 respectively.
Row 2 gives A001057 (canonical enumeration of integers).
Row 3 gives 0 together with A008795 (Molien series for 3-dimensional representation of dihedral group D_6 of order 6).
Row 4 gives A008794 (squares repeated) except the initial zero.
Finally, for n >= 5 row n gives the generalized k-gonal numbers (see Crossrefs section).

			Array begins:
------------------------------------------------------------------
n\m  Seq. No.    0   1  -1   2  -2   3   -3    4   -4    5   -5
------------------------------------------------------------------
0    A317300:    0,  1, -3,  0, -8, -3, -15,  -8, -24, -15, -35...
1    A317301:    0,  1, -2,  1, -5,  0,  -9,  -2, -14,  -5, -20...
2    A001057:    0,  1, -1,  2, -2,  3,  -3,   4,  -4,   5,  -5...
3   (A008795):   0,  1,  0,  3,  1,  6,   3,  10,   6,  15,  10...
4   (A008794):   0,  1,  1,  4,  4,  9,   9,  16,  16,  25,  25...
5    A001318:    0,  1,  2,  5,  7, 12,  15,  22,  26,  35,  40...
6    A000217:    0,  1,  3,  6, 10, 15,  21,  28,  36,  45,  55...
7    A085787:    0,  1,  4,  7, 13, 18,  27,  34,  46,  55,  70...
8    A001082:    0,  1,  5,  8, 16, 21,  33,  40,  56,  65,  85...
9    A118277:    0,  1,  6,  9, 19, 24,  39,  46,  66,  75, 100...
10   A074377:    0,  1,  7, 10, 22, 27,  45,  52,  76,  85, 115...
11   A195160:    0,  1,  8, 11, 25, 30,  51,  58,  86,  95, 130...
12   A195162:    0,  1,  9, 12, 28, 33,  57,  64,  96, 105, 145...
13   A195313:    0,  1, 10, 13, 31, 36,  63,  70, 106, 115, 160...
14   A195818:    0,  1, 11, 14, 34, 39,  69,  76, 116, 125, 175...
15   A277082:    0,  1, 12, 15, 37, 42,  75,  82, 126, 135, 190...
...

Alois P. Heinz, Antidiagonals n = 0..200 (first 45 antidiagonals from Robert G. Wilson v)

Columns 0..2 are A000004, A000012, A023445.
Column 3 gives A001477 which coincides with the row numbers.
Main diagonal gives A292551.
Row 0-2 gives A317300, A317301, A001057.
Row 3 gives 0 together with A008795.
Row 4 gives A008794.
For n >= 5, rows n gives the generalized n-gonal numbers: A001318 (n=5), A000217 (n=6), A085787 (n=7), A001082 (n=8), A118277 (n=9), A074377 (n=10), A195160 (n=11), A195162 (n=12), A195313 (n=13), A195818 (n=14), A277082 (n=15), A274978 (n=16), A303305 (n=17), A274979 (n=18), A303813 (n=19), A218864 (n=20), A303298 (n=21), A303299 (n=22), A303303 (n=23), A303814 (n=24), A303304 (n=25), A316724 (n=26), A316725 (n=27), A303812 (n=28), A303815 (n=29), A316729 (n=30).
Cf. A194801, A195152.
Cf. A317302 (a similar table but with polygonal numbers).

t[n_, r_] := PolygonalNumber[n, If[OddQ@ r, Floor[(r + 1)/2], -r/2]]; Table[ t[n - r, r], {n, 0, 11}, {r, 0, n}] // Flatten (* also *)
(* to view the square array *)  Table[ t[n, r], {n, 0, 15}, {r, 0, 10}] // TableForm (* Robert G. Wilson v, Aug 08 2018 *)

T(n,k) = A194801(n-3,k) if n >= 3.

A081056 Number of partitions of 2n+1 in which no parts are multiples of 4.

Original entry on oeis.org

1, 3, 6, 12, 22, 38, 64, 105, 166, 258, 395, 592, 876, 1280, 1846, 2636, 3728, 5222, 7256, 10006, 13696, 18624, 25169, 33808, 45164, 60022, 79366, 104457, 136870, 178572, 232044, 300368, 387366, 497804, 637568, 813910, 1035792, 1314214

Offset: 0

Michael Somos, Mar 03 2003

Euler transform of period 16 sequence [3,0,2,1,2,1,3,0,3,1,2,1,2,0,3,0,...].

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. A001935, A074377, A081055.

Table[Count[IntegerPartitions[2n+1],?(Total[Boole[Divisible[#,4]]]==0&)],{n,0,40}] (* _Harvey P. Dale, Sep 11 2019 *)

a(n)=local(X); if(n<0,0,X=x+x^2*O(x^(2*n)); polcoeff(eta(X^4)/eta(X),2*n+1))

G.f.: (sum_{n>=0} x^A074377(n))/(sum_n (-x)^n^2).
a(n) = A001935(2n+1).
a(n) ~ exp(Pi*sqrt(n)) / (2^(7/2) * n^(3/4)). - Vaclav Kotesovec, Nov 15 2017

A108211 a(n) = 16*n^2 + 1.

Original entry on oeis.org

17, 65, 145, 257, 401, 577, 785, 1025, 1297, 1601, 1937, 2305, 2705, 3137, 3601, 4097, 4625, 5185, 5777, 6401, 7057, 7745, 8465, 9217, 10001, 10817, 11665, 12545, 13457, 14401, 15377, 16385, 17425, 18497, 19601, 20737, 21905, 23105, 24337, 25601, 26897, 28225, 29585

Offset: 1

Reinhard Zumkeller, Jun 15 2005

Area of a Maltese cross conventionally inscribed in a 5n X 5n-grid.
Areas of some other crosses, each made from unit squares, as shown in Weisstein's illustrations: Greek Cross = x-pentomino = 5. Latin Cross = 6. Saint Andrew's cross = crux decussata = 9. Saint Anthony's Cross = tau cross = crux commissa = 10. Gaullist Cross = cross of Lorraine or patriarchal cross = 13. Papal Cross = 22. - Jonathan Vos Post, Jun 18 2005
The identity (16*n^2 + 1)^2 - (64*n^2 + 8)*(2*n)^2 = 1 can be written as a(n)^2 - A158488(n)*A005843(n)^2 = 1. - Vincenzo Librandi, Feb 08 2012
Sequence found by reading the line from 17, in the direction 17, 65, ... in the square spiral whose vertices are the generalized decagonal numbers A074377. - Omar E. Pol, Nov 02 2012
Conjecture: a(n) = floor(1/(1/(4*n) - log(2) + 1/(n+1) + 1/(n+2) + ... + 1/(2*n))). - Clark Kimberling, Sep 09 2014

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Maltese Cross.
Eric Weisstein's World of Mathematics, Gaullist Cross.
Eric Weisstein's World of Mathematics, Greek Cross.
Eric Weisstein's World of Mathematics, Latin Cross.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A002522, A005843, A016802, A053755, A074377, A158488.

I:=[17, 65, 145]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+Self(n-3): n in [1..50]]; // Vincenzo Librandi, Feb 08 2012

A108211:=n->16*n^2+1: seq(A108211(n), n=1..50); # Wesley Ivan Hurt, Sep 24 2014

LinearRecurrence[{3, -3, 1}, {17, 65, 145}, 40] (* Vincenzo Librandi, Feb 08 2012 *)
16*Range[50]^2+1 (* Harvey P. Dale, Jun 06 2025 *)

a(n)= 16*n^2+1 \\ Charles R Greathouse IV, Dec 23 2011

a(n) = A002522(4*n) = A016802(n) + 1.
G.f.: x*(17+14*x+x^2)/(1-x)^3. - Bruno Berselli, Feb 08 2012
From Amiram Eldar, Jul 13 2020: (Start)
Sum_{n>=1} 1/a(n) = Pi*coth(Pi/4)/8 - 1/2.
Sum_{n>=1} (-1)^(n+1)/a(n) = 1/2 - Pi*csch(Pi/4)/8. (End)
From Amiram Eldar, Feb 05 2021: (Start)
Product_{n>=0} (1 + 1/a(n)) = sqrt(2)*csch(Pi/4)*sinh(Pi/sqrt(8)).
Product_{n>=1} (1 - 1/a(n)) = (Pi/4)*csch(Pi/4). (End)
From Elmo R. Oliveira, Jan 17 2025: (Start)
E.g.f.: exp(x)*(16*x^2 + 16*x + 1) - 1.
a(n) = A053755(2*n).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 3. (End)

A143838 Ulam's spiral (SSW spoke).

Original entry on oeis.org

1, 22, 75, 160, 277, 426, 607, 820, 1065, 1342, 1651, 1992, 2365, 2770, 3207, 3676, 4177, 4710, 5275, 5872, 6501, 7162, 7855, 8580, 9337, 10126, 10947, 11800, 12685, 13602, 14551, 15532, 16545, 17590, 18667, 19776, 20917, 22090, 23295, 24532

Offset: 1

Vladimir Joseph Stephan Orlovsky, Sep 02 2008

Also sequence found by reading the segment (1, 22) together with the line from 22, in the direction 22, 75, ..., in the square spiral whose vertices are the generalized decagonal numbers A074377. - Omar E. Pol, Nov 05 2012

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

List([1..50], n-> ((32*n-27)^2 +39)/64); # G. C. Greubel, Nov 09 2019

[16*n^2-27*n+12: n in [1..50]]; // Vincenzo Librandi, Nov 08 2014

seq( ((32*n-27)^2 +39)/64, n=1..50); # G. C. Greubel, Nov 09 2019

f[n_]:= 16n^2 -27n +12; Array[f, 40] (* Vladimir Joseph Stephan Orlovsky, Sep 02 2008 *)
CoefficientList[Series[(1+19x+12x^2)/(1-x)^3, {x, 0, 40}], x] (* Vincenzo Librandi, Nov 08 2014 *)
((32*Range[50]-27)^2 +39)/64 (* G. C. Greubel, Nov 09 2019 *)
LinearRecurrence[{3,-3,1},{1,22,75},40] (* Harvey P. Dale, Sep 26 2020 *)

vector(50, n, 16*n^2-27*n+12) \\ Michel Marcus, Nov 08 2014

[((32*n-27)^2 +39)/64 for n in (1..50)] # G. C. Greubel, Nov 09 2019

a(n) = 16*n^2 - 27*n + 12, n>0. - R. J. Mathar, Sep 04 2008
G.f.: x*(1 + 19*x + 12*x^2)/(1-x)^3. - Colin Barker, Aug 03 2012
E.g.f.: -12 + (12 - 11*x + 16*x^2)*exp(x). - G. C. Greubel, Nov 09 2019

A143839 Ulam's spiral (SSE spoke).

Original entry on oeis.org

1, 24, 79, 166, 285, 436, 619, 834, 1081, 1360, 1671, 2014, 2389, 2796, 3235, 3706, 4209, 4744, 5311, 5910, 6541, 7204, 7899, 8626, 9385, 10176, 10999, 11854, 12741, 13660, 14611, 15594, 16609, 17656, 18735, 19846, 20989, 22164, 23371, 24610

Offset: 1

Vladimir Joseph Stephan Orlovsky, Sep 02 2008

Also sequence found by reading the line from 1, in the direction 1, 24, ... in the square spiral whose vertices are the generalized decagonal numbers A074377. - Omar E. Pol, Nov 05 2012

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A004767, A016813.

List([1..40], n-> ((32*n -25)^2 +15)/64); # G. C. Greubel, Nov 09 2019

[16*n^2-25*n+10: n in [1..40]]; // Vincenzo Librandi, Nov 08 2014

seq( ((32*n -25)^2 +15)/64, n=1..40); # G. C. Greubel, Nov 09 2019

f[n_] := 16n^2 -25n +10; Array[f, 40] (* Vladimir Joseph Stephan Orlovsky, Sep 02 2008 *)
LinearRecurrence[{3,-3,1},{1,24,79},40] (* Harvey P. Dale, May 26 2012 *)
CoefficientList[Series[(1+21*x+10*x^2)/(1-x)^3, {x, 0, 40}], x] (* Vincenzo Librandi, Nov 08 2014 *)

Vec(x*(1+21*x+10*x^2)/(1-x)^3 + O(x^40)) \\ Colin Barker, Nov 08 2014

[((32*n -25)^2 +15)/64 for n in (1..40)] # G. C. Greubel, Nov 09 2019

a(n) = 16*n^2 - 25*n + 10. - R. J. Mathar, Sep 04 2008
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3); a(1) = 1, a(2) = 24, a(3) = 79. - Harvey P. Dale, May 26 2012
G.f.: x*(1 + 21*x + 10*x^2)/(1-x)^3. - Harvey P. Dale, May 26 2012
E.g.f.: exp(x)*(10 - 9*x + 16*x^2) - 9. - Stefano Spezia, Oct 07 2019

A143854 Ulam's spiral (WSW spoke).

Original entry on oeis.org

1, 20, 71, 154, 269, 416, 595, 806, 1049, 1324, 1631, 1970, 2341, 2744, 3179, 3646, 4145, 4676, 5239, 5834, 6461, 7120, 7811, 8534, 9289, 10076, 10895, 11746, 12629, 13544, 14491, 15470, 16481, 17524, 18599, 19706, 20845, 22016, 23219, 24454

Offset: 1

Vladimir Joseph Stephan Orlovsky, Sep 03 2008

Also sequence found by reading the segment (1, 20) together with the line from 20, in the direction 20, 71, ..., in the square spiral whose vertices are the generalized decagonal numbers A074377. - Omar E. Pol, Nov 05 2012

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

List([1..50], n-> ((32*n-29)^2 +55)/64); # G. C. Greubel, Nov 09 2019

[14-29*n+16*n^2: n in [1..50]]; // Vincenzo Librandi, Nov 08 2014

seq( ((32*n-29)^2 +55)/64, n=1..40); # G. C. Greubel, Nov 09 2019

f[n_]:= 16n^2 -29n +14; Array[f, 40] (* Vladimir Joseph Stephan Orlovsky, Sep 03 2008 *)
CoefficientList[Series[(1+17x+14x^2)/(1-x)^3, {x, 0, 40}], x] (* Vincenzo Librandi, Nov 08 2014 *)
((32*Range[50] -29)^2 +55)/64 (* G. C. Greubel, Nov 09 2019 *)

vector(50, n, 14-29*n+16*n^2) \\ Michel Marcus, Nov 08 2014

[((32*n-29)^2 +55)/64 for n in (1..50)] # G. C. Greubel, Nov 09 2019

From Colin Barker, Aug 03 2012: (Start)
a(n) = 14 - 29*n + 16*n^2.
G.f.: x*(1 + 17*x + 14*x^2)/(1-x)^3. (End)
E.g.f.: -14 + (14 - 13*x + 16*x^2)*exp(x). - G. C. Greubel, Nov 09 2019

A143855 Ulam's spiral (ESE spoke).

Original entry on oeis.org

1, 10, 51, 124, 229, 366, 535, 736, 969, 1234, 1531, 1860, 2221, 2614, 3039, 3496, 3985, 4506, 5059, 5644, 6261, 6910, 7591, 8304, 9049, 9826, 10635, 11476, 12349, 13254, 14191, 15160, 16161, 17194, 18259, 19356, 20485, 21646, 22839, 24064

Offset: 1

Vladimir Joseph Stephan Orlovsky, Sep 03 2008, Sep 04 2008

Also sequence found by reading the segment (1, 10) together with the line from 10, in the direction 10, 51, ..., in the square spiral whose vertices are the generalized decagonal numbers A074377. - Omar E. Pol, Nov 05 2012

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

List([1..50], n-> ((32*n-39)^2 +15)/64); # G. C. Greubel, Nov 09 2019

[16*n^2-39*n+24: n in [1..50]]; // Vincenzo Librandi, Nov 08 2014

seq( ((32*n -39)^2 +15)/64, n=1..50); # G. C. Greubel, Nov 09 2019

f[n_]:= 16n^2 -39n +24; Array[f, 40] (* Vladimir Joseph Stephan Orlovsky, Sep 04 2008 *)
CoefficientList[Series[(1+7x+24x^2)/(1-x)^3, {x,0,40}], x] (* Vincenzo Librandi, Nov 08 2014 *)
((32*Range[50] -39)^2 +15)/64 (* G. C. Greubel, Nov 09 2019 *)

vector(50, n, 16*n^2-39*n+24) \\ Michel Marcus, Nov 08 2014

[((32*n-39)^2 +15)/64 for n in (1..50)] # G. C. Greubel, Nov 09 2019

a(n) = 16*n^2 - 39*n + 24. - R. J. Mathar, Sep 08 2008
G.f.: x*(1 + 7*x + 24*x^2)/(1-x)^3. - Colin Barker, Aug 03 2012
E.g.f.: -24 + (24 - 23*x + 16*x^2)*exp(x). - G. C. Greubel, Nov 09 2019

A143856 Ulam's spiral (ENE spoke).

Original entry on oeis.org

1, 12, 55, 130, 237, 376, 547, 750, 985, 1252, 1551, 1882, 2245, 2640, 3067, 3526, 4017, 4540, 5095, 5682, 6301, 6952, 7635, 8350, 9097, 9876, 10687, 11530, 12405, 13312, 14251, 15222, 16225, 17260, 18327, 19426, 20557, 21720, 22915, 24142

Offset: 1

Vladimir Joseph Stephan Orlovsky, Sep 03 2008

Also sequence found by reading the segment (1, 12) together with the line from 12, in the direction 12, 55,..., in the square spiral whose vertices are the generalized decagonal numbers A074377. - Omar E. Pol, Nov 05 2012

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

List([1..40], n-> ((32*n-37)^2 +39)/64); # G. C. Greubel, Nov 09 2019

[16*n^2 -37*n +22: n in [1..40]]; // Vincenzo Librandi, Jul 10 2012

seq( ((32*n-37)^2 +39)/64, n=1..40); # G. C. Greubel, Nov 09 2019

f[n_]:= 16n^2 -37n +22; Array[f, 40] (* Robert G. Wilson v, Oct 31 2011 *)
Table[16n^2-37*n+22,{n,1,40}] (* Vincenzo Librandi, Jul 10 2012 *)
LinearRecurrence[{3,-3,1},{1,12,55},50] (* Harvey P. Dale, Sep 02 2024 *)

a(n)=16*n^2-37*n+22 \\ Charles R Greathouse IV, Jun 17 2017

[((32*n-37)^2 +39)/64 for n in (1..40)] # G. C. Greubel, Nov 09 2019

a(n) = 16*n^2 - 37*n + 22. - R. J. Mathar, Sep 08 2008
G.f. x*(1 + 9*x + 22*x^2)/(1-x)^3. - R. J. Mathar, Oct 31 2011
a(n) = 3*a(n-1) -3*a(n-2) +a(n-3). - Vincenzo Librandi, Jul 10 2012
E.g.f.: -22 + (22 - 21*x + 16*x^2)*exp(x). - G. C. Greubel, Nov 09 2019

cp's OEIS Frontend

A274757 Numbers k such that 6*k+1 is a triangular number (A000217).

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A274830 Numbers k such that 7*k+1 is a triangular number (A000217).

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A303301 Square array T(n,k) read by antidiagonals upwards in which row n is obtained by taking the general formula for generalized n-gonal numbers: m*((n - 2)*m - n + 4)/2, where m = 0, +1, -1, +2, -2, +3, -3, ... and n >= 5. Here n >= 0.

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A081056 Number of partitions of 2n+1 in which no parts are multiples of 4.

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PARI

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A108211 a(n) = 16*n^2 + 1.

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Magma

Maple

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A143838 Ulam's spiral (SSW spoke).

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A143839 Ulam's spiral (SSE spoke).

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A303301 Square array T(n,k) read by antidiagonals upwards in which row n is obtained by taking the general formula for generalized n-gonal numbers: m((n - 2)m - n + 4)/2, where m = 0, +1, -1, +2, -2, +3, -3, ... and n >= 5. Here n >= 0.